HJB equations for stochastic control problems with delay in the control: regularity and feedback controls

Stochastic optimal control problems governed by delay equations
with delay in the control are usually more difficult to study than the ones
when the delay appears only in the state. This is particularly true when we
look at the associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even
in the simplified setting (introduced first by Vinter and Kwong for the
deterministic case) the HJB equation is an infinite dimensional second
order semi-linear PDE that does not satisfy the so-called structure
condition which substantially means that "the noise enters the system with
the control". The absence of such condition, together with the lack
of smoothing properties which is a common feature of problems with delay,
prevents the use of known techniques (based on Backward Stochastic
Differential Equations or on the smoothing properties of the linear part)
to prove the existence of regular solutions to this HJB equation and thus
no results in this direction have been proved till now. In this talk we
will discuss results about existence of regular solutions of this kind of
HJB equations and their use in solving the
corresponding control problem by finding optimal feedback controls,
also in the more difficult case of pointwise delay.
This is a joint work with Federica Masiero.